Stan-Based Epidemiological Models II: Modeling Dengue

dS_h/dt = -(b β_h I_m S_h)/N_h
dE_h/dt = (b β_h I_m S_h)/N_h - k_h E_h
dI_h/dt = k_h E_h - γ_h I_h
dR_h/dt = γ_h I_h

The infection term (b β_h I_m S_h)/N_h represents the rate at which susceptible humans become infected through mosquito bites, and it captures the fundamental mechanics of dengue transmission. This term depends on several key factors working together: the number of infectious mosquitoes in the population (I_m), the biting rate or how frequently each mosquito bites humans per day (b), the probability that a bite from an infectious mosquito actually transmits the virus to a susceptible person (β_h), and the number of susceptible people available to be bitten (S_h). The term is divided by the total human population (N_h) to convert this into a proper rate per capita. Essentially, this tells us that disease transmission increases when there are more infectious mosquitoes around, when mosquitoes bite more frequently, when each bite is more likely to cause infection, and when there are more susceptible people in the population. It's scaled by population size so that larger populations don't automatically have higher per-person infection rates.

The exposed humans equation dE_h/dt = (b β_h I_m S_h)/N_h - k_h E_h describes how people flow into and out of the incubation stage, where the inflow is exactly the infection term from susceptible humans (people who just got bitten and infected), while the outflow k_h E_h represents exposed people finishing their incubation period and becoming infectious at rate k_h. The infectious humans equation dI_h/dt = k_h E_h - γ_h I_h shows that people become infectious after incubation (inflow k_h E_h) and then recover at rate γ_h, making γ_h I_h the number of people recovering per day. Finally, the recovered equation dR_h/dt = γ_h I_h simply captures that all the people recovering from infection (γ_h I_h) join the imμne population, with no outflow since recovered individuals remain imμne to that dengue strain.

The mosquito model is simpler: they don't recover, just die. But we have another term here, birth and death rates of mosquitoes.

dS_m/dt = μ_m N_m - (b β_m I_h S_m)/N_h - μ_m S_m
dE_m/dt = (b β_m I_h S_m)/N_h - k_m E_m - μ_m E_m
dI_m/dt = k_m E_m - μ_m I_m

The mosquito population dynamics follow a simpler SEI pattern since mosquitoes don't recover from dengue infection. The susceptible mosquito equation dS_m/dt = μ_m N_m - (b β_m I_h S_m)/N_h - μ_m S_m shows that new mosquitoes are constantly born at rate μ_m N_m (where μ_m is the mosquito mortality rate of about 0.134-0.164 per day, corresponding to their short 14-28 day lifespan), while susceptible mosquitoes leave this compartment either by getting infected when they bite infectious humans at rate (b β_m I_h S_m)/N_h (where b is the biting rate of ~1 bite per day, β_m is the human-to-mosquito transmission probability of 0.3-0.9, and I_h is the number of infectious humans) or by dying at rate μ_m S_m. The exposed mosquito equation dE_m/dt = (b β_m I_h S_m)/N_h - k_m E_m - μ_m E_m captures mosquitoes that have acquired the virus but are still in the extrinsic incubation period, flowing in from new infections and flowing out either by completing incubation at rate k_m (approximately 1/10 per day for the standard 8-12 day incubation period) or by dying. Finally, the infectious mosquito equation dI_m/dt = k_m E_m - μ_m I_m represents mosquitoes that have completed viral development and can now transmit dengue to humans, with inflow from completed incubation and outflow only through death since mosquitoes remain infectious for life.

But why did we not include births and deaths in the human SEIR model. For short-term dengue outbreaks lasting weeks to months, it would not make sense to include human births and deaths because these demographic terms (μ_h, μ_m) are often negligible compared to disease dynamics. Consider the timescales involved: human lifespan averages around 70 years, giving a daily mortality rate μ_h of approximately 0.00004 per day, while dengue incubation takes about 5.5 days (k_h ≈ 0.18/day) and recovery occurs in roughly 6 days (γ_h ≈ 0.17/day). This means the disease processes operate about 4,000 times faster than human demographics, making births and deaths essentially irrelevant noise when studying acute outbreaks where the epidemic dynamics unfold over weeks rather than years, as in this example case.